The Loop Algorithm

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  • 11 January 2000
Abstract
A review of the Loop Algorithm and its generalizations is given, including new results. The loop algorithm is a Monte Carlo procedure which performs nonlocal changes of worldline configurations, determined by local stochastic decisions. It is based on a formulation of physical models in an extended ensemble of worldlines and graphs, and is related to Swendsen-Wang cluster algorithms. It overcomes many of the difficulties of traditional worldline simulations. Autocorrelations between successive Monte Carlo configurations are reduced by orders of magnitude. The grand-canonical ensemble (e.g. varying winding numbers) is naturally simulated. The continuous time limit can be taken directly. Off-diagonal operators can be measured. Improved Estimators exist which further reduce the errors of measured quantities. For a large class of models, the fermion sign problem can be overcome. The algorithm remains unchanged in any dimension and for varying bond-strengths. It becomes less efficient in the presence of strong site disorder or strong diagonal magnetic fields. Transverse fields are handled efficiently. It applies directly to locally XYZ-like spin, fermion, and hard-core boson models. It has been extended to the Hubbard and the tJ model and generalized to higher spin representations. It has been used in many large scale applications, especially for Heisenberg-like models, where lattices of a million sites and temperatures below 0.001J have become accessible, allowing the precise calculation of asymptotic behavior and of quantum critical exponents.

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